We have made high time-resolution polarimetric observations of the
37.9\,ms relativistic binary pulsar B1534+12. Roughly 20\% of the main
pulse emission is linearly polarized, and the position-angle swing
within the pulse shows structure clearly inconsistent with purely
dipolar emission. Our measurements were also sensitive to low-level
off-pulse emission, allowing us to trace the variation in polarization
angle, globally dipolar in nature, through a substantial fraction
of the pulsar's rotation period. A preliminary rotating vector model
fit to the position-angle swing results in an impact parameter (the
angle between the magnetic axis of the pulsar and our line of sight) of
$\beta = (3.5 \pm 1.5)^\circ$ and a magnetic alignment (the angle
between the spin and magnetic axes of the pulsar) of $\alpha = (106 \pm
8)^\circ$. In addition to providing the geometry of the system and
clues to the pulsar's emission mechanism, these measurements form a
pulse polarization ``template'' against which future observations can
be compared in order to constrain or detect the effects of relativistic
precession of the pulsar's spin axis in its binary orbit.